The effects of probabilistic innovations on Schumpeterian economic growth in a creative region

Economic Modelling 53 (2016) 224–230

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The effects of probabilistic innovations on Schumpeterian economic growth in a creative region☆ Amitrajeet A. Batabyal a,⁎, Hamid Beladi b a b

Department of Economics, Rochester Institute of Technology, 92 Lomb Memorial Drive, Rochester, NY 14623-5604, USA Department of Economics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249-0631, USA

a r t i c l e

i n f o

Article history: Accepted 20 November 2015 Available online 30 December 2015 Keywords: Creative capital Creative region Innovation Schumpeterian economic growth Uncertainty

a b s t r a c t We analyze the impact that stochastically occurring innovations have on Schumpeterian economic growth in a region that is creative in the sense of Richard Florida. Our analysis leads to four findings. First, we delineate the so called balanced growth path (BGP) equilibrium and then compute the BGP growth rate in our creative region. Second, we discuss why the lower limit of the support of the random variable that describes the outcome of innovation quality improvements, takes the value that it does. Third, we solve the social planner's problem and derive the Pareto optimal growth rate in our creative region. Finally, we compare the BGP and the Pareto optimal growth rates, we discuss when there is either too much or too little innovation, and we conclude by commenting on the implications of our findings for future research on Schumpeterian economic growth in creative regions. © 2015 Elsevier B.V. All rights reserved.

1. Introduction 1.1. Objective and rationale A result of the academic writings and the popular musings of the urbanist Richard Florida1 is that economists and regional scientists are now very familiar with the two related notions of the creative class and creative capital. Florida (2002, p. 68) helpfully explains that the creative class “consists of people who add economic value through their creativity.” This class is composed of professionals such as doctors, lawyers, scientists, engineers, university professors, and, notably, bohemians such as artists, musicians, and sculptors. From the standpoint of regional economic growth and development, these people are important because they possess creative capital which is the “intrinsically human ability to create new ideas, new technologies, new business models, new cultural forms, and whole new industries that really [matter]” (Florida, 2005, p. 32). As pointed out by Florida on a number of occasions, the creative class is noteworthy because this group gives rise to ideas, information, and technology, outputs that are important for the growth and development of cities and regions. Therefore, in this era of globalization, cities and regions that want to be successful need to do all they can to attract and retain members of the creative class because this class is the principal driver of economic growth. ☆ We thank the Editor Sushanta Mallick and three anonymous referees for their helpful comments on two previous versions of this paper. In addition, Batabyal acknowledges sabbatical funding from RIT and financial support from the Gosnell endowment, also at RIT. The usual disclaimer applies. ⁎ Corresponding author. Tel.: +1 585 475 2805; fax: +1 585 475 5777. E-mail addresses: [email protected] (A.A. Batabyal), [email protected] (H. Beladi). 1 See Florida (2002, 2005) and Florida et al. (2008).

http://dx.doi.org/10.1016/j.econmod.2015.11.026 0264-9993/© 2015 Elsevier B.V. All rights reserved.

How is the concept of creative capital different from the more familiar and traditional notion of human capital? To answer this question, first note that in empirical work, the notion of human capital is generally measured with education or with education based indicators. This notwithstanding, Marlet and Van Woerkens (2007) have rightly noted that the accumulation of creative capital does not have to be a function of the acquisition of formal education. What this means is that although the creative capital accumulated by some members of Florida's creative class (doctors, engineers, university professors) is a function of the completion of many years of formal education, the same is not always true of other members of this creative class (artists, painters, poets). Individuals in this latter group may be innately creative and hence possess raw creative capital despite having very little or no formal education. Given this state of affairs, we agree with Marlet and Van Woerkens (2007) and assert that there is little or no difference between the notions of human and creative capital when the accumulation of this creative capital depends on the completion of many years of formal education. In contrast, there can be a lot of difference between the notions of human and creative capital when the accumulation of this creative capital does not have to depend on the completion of formal education. Since creative capital is of two types, it is a more general concept than the more traditional notion of human capital. With this explanatory background in place, let us now emphasize three points. First, Eversole (2005), Siemiatycki (2013), and others have noted that in regions where the creative class is a dominant part of the overall workforce, there is a definite link between innovations and the activities of the members of this creative class. In this regard, Fischer and Nijkamp (2009), Baumol (2010), and Batabyal and Nijkamp (2013) have stressed that innovation is an important driver of regional economic growth and development.

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Second, the existing literature has recognized that innovative activities and processes are fundamentally competitive in nature and that this competitive aspect is closely related to the critical insight of Joseph Schumpeter who argued that growth processes are characterized by creative destruction in which “economic growth is driven, at least in part, by new firms replacing incumbents and new machines and products replacing old ones” (Acemoglu, 2009, p. 458). Finally, the preceding two points notwithstanding, the reader should understand that there are no theoretical studies of Schumpeterian economic growth in a region that is creative in the sense of Richard Florida. Therefore, in this paper, we provide the first theoretical analysis of the effects that probabilistic or stochastic innovations have on Schumpeterian economic growth in a region that is creative a la Richard Florida. Now, before we proceed to the specifics of our paper, we first briefly survey the related literature on Schumpeterian economic growth. 1.2. Review of the literature Without concentrating on creative regions per se, Leahy and McKee (1972) stated but did not model the idea that change in regional economies can be appropriately understood by adopting a “Schumpeterian view” of the regional economy. In spite of the appearance of this statement more than four decades ago, economists and regional scientists have begun to use the ideas of Schumpeter to investigate the nexus between innovation and economic growth in generic regions only since the early 1980s. As such, there is now a fairly substantial empirical and case study based literature that has analyzed several different aspects of Schumpeterian economic growth in generic regional economies. At the regional level, Van Wissen and Huisman (2002) show that large growth differentials are likely to be observed between what they call “core,” “ring,” and “peripheral” regions. Lodde (2008) demonstrates that there is qualified support for the Schumpeterian hypothesis in selected regions in Italy. Crespi and Pianta (2008) concentrate on six nations in Europe and point out that the ideas of Schumpeter are useful in comprehending the variety of innovation and what they call “innovation–performance relationships” in the six countries under study. Quatraro (2009) contends that Schumpeter's views about innovation and business cycles can be used to comprehend the diffusion of innovation capabilities in various Italian regions. Aghion et al. (2009) point out that there is empirical support for the idea that more intense competition enhances innovation among what they call “frontier” firms but that this kind of intense competition may actually discourage innovation in “non-frontier” firms. Focusing on innovative firms, Akcigit and Kerr (2010) and Haltiwanger et al. (2010) point out that firm size and age are positively correlated and that in innovative industries, small firms exit the industry more often and hence the surviving firms tend to grow relatively rapidly. Concentrating on 2,645 counties in the United States, Hodges and Ostbye (2010) find support for a Schumpeterian growth model because, in their empirical model, bigger firms are needed to carry out effective R&D which then leads to higher economic growth in the localities being studied. Saunoris and Payne (2011) use United States data from 1960 to 2007 and demonstrate that long run increases in R&D expenditures are necessary to offset lower R&D productivity due to the presence of product proliferation. Shen (2014) shows that the impact of wealth inequality on Schumpeterian economic growth arises through the supply of human capital as well as the demand for higher quality goods. Finally, Carillo and Papagni (2014) utilize a Schumpeterian growth model and make the point that the incentive structure confronting an economy's science sector greatly influences both the development of science and the economy itself. There are only two theoretical studies that are loosely connected to the basic question we study—see Section 1.1—in this paper. Batabyal and Nijkamp (2012) have analyzed a one-sector, discrete-time,

225

Schumpeterian model of growth in a general region that is not necessarily creative. They show that the region being studied experiences bursts of unemployment followed by periods of full employment. There is no overlap between the question studied in this paper and our present paper. More recently, Batabyal and Nijkmap (2014) have used a Schumpeterian growth model to study a question similar to the one we study here. However, there are three key differences between this paper and our present paper. Specifically, Batabyal and Nijkamp (2014) focus on generic regions, innovations reduce the marginal cost of producing machines or intermediate goods, and the impact of the innovations is deterministic. In contrast, our paper focuses on a creative region, innovations improve the quality of existing machines, and the effect of innovations is probabilistic. The remainder of this paper is arranged as follows. Recall that our basic goal here is to analyze the effects of probabilistic innovations on Schumpeterian economic growth in a creative region. To this end, Section 2 describes our theoretical model of a creative region that is adapted from Aghion and Howitt (1992) and Acemoglu (2009, pp. 459–472). Section 3 describes the so called balanced growth path (BGP) equilibrium and then computes the BGP growth rate in our creative region. Section 4 discusses why the lower limit of the support of the random variable that describes the effect of innovation quality improvements, takes the value that it does. Section 5 solves the social planner's problem and then derives the Pareto optimal growth rate in our creative region. Section 6 compares the BGP and the Pareto optimal growth rates and then discusses when there is either too much or too little innovation in our creative region. Finally, Section 7 concludes and then offers three suggestions for extending the research described here. 2. The theoretical framework 2.1. Preliminaries Consider an infinite horizon, stylized region that is creative in the sense of Richard Florida. This creative region is made up of J distinct spatial units which we index with the superscript j where j = 1, 2, …, J. The representative creative class household in our region displays constant relative risk aversion (CRRA) and its CRRA utility function is denoted 1‐θ − 1}/(1 − θ)]dt, θ ≠ 1, where C(t) is consumpby ∫∞ 0 exp(−ρt)[{C(t) tion in time t, ρ N 0 is the constant time discount rate, and θ ≥ 0 is the constant coefficient of relative risk aversion.2 The creative region under study possesses creative capital in each of its J distinct spatial units. The creative capital in the jth spatial unit at time t is denoted by Rj(t). Clearly, the total stock of creative capital in our region at any time t is given by R(t) = ∑Jj = 1Rj(t). There is no growth in the stock of creative capital R(t) and this R(t) is supplied inelastically. The aggregate resource or budget constraint in our region at time t is given by C ðt Þ þ X ðt Þ þ Iðt Þ ≤ Oðt Þ;

ð1Þ

where C(t) is consumption, X(t) is total spending on machines, I(t) is total spending on R&D, and O(t) is the output of the competitively produced single final good for consumption that we shall think of as a knowledge good such as a smartphone or a desktop computer. The price of this final good is normalized to unity at all points in time and hence O(t) denotes both the output and the value of the final good. In addition, note that the machines we have just referred to can also be thought of as inputs or as intermediate goods. The aggregate or total expenditure on R&D in our creative region or I(t) is the sum of the R&D expenditures incurred by each one of the J distinct spatial units in the region under study. In symbols, this means that I(t) = ∑Jj = 1I j(t). 2 See Acemoglu (2009, pp. 308–309) for additional details on the properties of the CRRA utility function.

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There is a continuum of machines that is used to produce the single final good O(t). Each machine line or variety3 is described by v where v ϵ [0,1]. The source of economic growth in our creative region is process innovations that improve the quality of existing machines. To this end, let q(v,t) be the quality of the machine of line v at time t. We assume that conditional on success, an innovation gives rise to a probabilistic improvement of α over the previous technology in use. Let G(α) denote the cumulative distribution function of α and let the support ^ 4 Note that β in this of α be given by the interval ½ð1−βÞ−ð1−βÞ=β ; α support is a parameter of the function describing the production of the final consumption good and is specified in greater detail in the next paragraph. As pointed out previously, the single final good for consumption (the knowledge good) in our creative region or O(t) is produced competitively and in a single location. The production function is 21 3 Z 1 4 1−β qðv; t Þxðv; t=qÞ dv5Rβ ; Oðt Þ ¼ 1−β

ð2Þ

0

where R is the creative capital input, q(v,t) is the quality of a machine of line v at time t, x(v,t/q) is the total amount of the machine of variety v and quality q that is used at time t and β ϵ (0,1) is the parameter of the production function that we have alluded to in the previous paragraph. Let w denote the wage paid to the creative capital input R and let r denote the interest rate. It is clear that R(t) = ∑Jj = 1Rj(t) which means that each spatial unit in our creative region supplies its own creative capital inelastically and thereby contributes additively and positively to the production of the final consumption good O(t) (see Eq. (2)). Because the final consumption good is produced in a single location, the creative capital units in each of the J spatial units travel over space to that location. The sum of these traveling creative capital units in this single location is denoted by R(t), which produces the final consumption good. This is one way in which the individual spatial units contribute to our analysis and thus there is one kind of spatial dimension in our analysis. Notice that for a given machine line v, only the machine with the highest or cutting edge quality is used to produce the single final good in equilibrium. This feature of the model is the source of creative destruction in the sense of Joseph Schumpeter. What this means is that when a machine with a higher quality is invented, it replaces or destroys the previous lower quality machine of the same line. Our next task is to explain how new machines in our creative region are first invented and then produced, Arrow's (1962) replacement effect and the related profit stealing effect, and finally, the appropriability effect. 2.2. Machine invention, production, replacement, profit stealing, and appropriability effects In our model, new machine varieties are invented by R&D and this R&D builds on the knowledge about existing machines. More specifically, the R&D in our creative region gives rise to innovation and this innovation advances the existing knowledge about the various machine lines. There is free entry into R&D. Therefore, any firm can carry out research on the existing machine lines. In particular, if a firm spends I(v,t) units of the final consumption good on R&D on a machine variety then this action gives rise to a flow rate of innovation given by ηI(v,t)/q(v,t), where η N 0 is a parameter and q(v,t) is the quality of the machine of variety v at time t. 3 We shall use the words “line” and “variety” interchangeably in the remainder of this paper. 4 In Section 4, we discuss why the lower limit of the support of the random variable α takes the specific value that it does.

The firm that makes an innovation receives a perpetual patent on the new machine it has invented. As such, this successful innovator has monopoly power in the market for machines.5 The cost of undertaking R&D is assumed to be the same for the incumbent monopolist and for new firms (potential entrants). The existing patent system does not preclude other firms from undertaking research based on the newly invented machine just mentioned. Note that in contrast to the centralized production of the final consumption good in our creative region, the invention and production of machines in this region is decentralized and therefore can occur in any of the J spatial units. In the model of this paper, the identity of the firm that conducts R&D is important. This is because as noted in Section 2.1, the quality of existing machines can be improved upon and it is this “quality improvement” process of creative destruction that is the source of economic growth in the creative region under study. This brings us to Arrow's (1962) prominent replacement effect. Since the cost of undertaking R&D is the same for the incumbent monopolist and for potential entrants, following the seminal work of Arrow, Acemoglu (2009, p. 460, emphasis added) has rightly pointed out that the “incumbent has weaker incentives to innovate, since it would replace its own machine (thus destroying the profit that it is already making).” In contrast, a potential entrant has no similar replacement computation to make and hence, given that the cost of undertaking R&D is the same for the incumbent monopolist and a potential entrant, it is always the entrants who carry out R&D. In addition, note that by replacing the incumbent monopolist, an entrant is also stealing this monopolist's profit. This is the profit stealing effect. In the model of Schumpeterian economic growth that we are studying in this paper, the social value of an innovation is always greater than the private value. This is because a social planner who would like to maximize the social surplus in our creative region will always be willing to adopt an innovation because this planner can appropriate the full value of the innovation. This is the appropriability effect. In contrast, an innovating firm that has monopoly rights in the aftermath of a successful innovation will be able to appropriate only a portion of the gain in the surplus created by the better technology. Put differently, the gain in social surplus with ex post monopoly rights is smaller than the social surplus gain that a planner can achieve.6 The reader should note that a key objective of ours in this paper—which we undertake in Section 6—is to compare the BGP and the Pareto optimal growth rates and then discuss when there is either too much or too little innovation in our creative region. Now, with this theoretical framework in place, our next task is to delineate the so called balanced growth path (BGP) equilibrium and then compute the BGP growth rate in our creative region. While undertaking this exercise, we shall adapt some of the results in Peters and Simsek (2009, pp. 171–179) to our analysis of Schumpeterian economic growth in a creative region.

3. The BGP equilibrium and growth rate We begin with some notation. Let p(v,t/q) and x(v,t/q) denote the price and the quantity of machine x of variety v and quality q at time t. Let the function V(v,t/q) denote the time t net present discounted value of the profit from a machine of variety v and quality q. Finally, 5 Note though that the value of a patent is independent of the machine variety v and time t. 6 The assumptions that we have made in Sections 2.1 and 2.2 about the consumption and the production sides of the economy of our creative region are standard and their rationale and empirical plausibility have been discussed in many growth theory and macroeconomics textbooks such as Acemoglu (2009) and Romer (2012). For additional details on these matters, see chapters 8, 12, 13, and 14 in Acemoglu (2009) and chapter 3 in Romer (2012). Readers wishing to see assumptions similar to those made in Sections 2.1 and 2.2 being employed in papers on economic growth with a significant empirical content should consult Acs and Audretsch (1988), Klette and Kortum (2004), and particularly Lentz and Mortensen (2008).

A.A. Batabyal, H. Beladi / Economic Modelling 53 (2016) 224–230

let i(v,t/q) denote the flow rate at which new innovations occur for machines of variety v. Now, the BGP equilibrium in our creative region is a collection of the trajectories of (i) consumption levels, total spending on machines, and total R&D expenditures [C(t), X(t), I(t)]∞ t = 0, (ii) qualities of the cutting edge machines [q(v,t)]∞ {0 ≤ v ≤ 1},t = 0, (iii) prices and quantities of machines and the net present discounted value of the profit from a machine [p(v, t/q), x(v, t/q), V(v, t/q)]∞ {0 ≤ v ≤ 1},t = 0, and (iv) interest and wage rates [r(t), w(t)] such that the following six conditions hold. First, the representative creative class household maximizes utility. Second, competitive producers of the single final good maximize their profit taking the price of the final good as given. Third, the monopolistic machine producers maximize their profit. Fourth, there is free entry into R&D. Fifth, aggregate output of the single final good {O(t)}and consumption {C(t)}grow at a common rate denoted by gB where the superscript B denotes the BGP. Finally, all markets clear. The discussion in the first paragraph of Appendix A tells us two things. First, the equilibrium profit of a machine producer with quality q(v,t) is given by π(v, t/q) = βq(v,t)R. Second, the value function V(v, t/q) described in the preceding paragraph solves a so called Hamilton– Jacobi–Bellman equation given by r ðt ÞV ðv; t=qÞ−V_ ðv; t=qÞ ¼ πðv; t=qÞ−iðv; t=qÞV ðv; t=qÞ;

ð3Þ

_ where Vðv; t=qÞ is the time derivative of the V(v, t/q) function. Two aspects of Eq. (3) deserve more commentary. First, the last term i(v, t/q)V(v, t/q) in this equation captures the essence of Schumpeterian economic growth because when a new innovation occurs, an existing monopolist loses its monopoly position and is therefore replaced by the producer of a higher quality machine. Second, the replacement effect—described in Section 2.2—is at work here because the innovation is undertaken by an entrant and not by the incumbent monopolist. Hence, it makes sense to think of i(v,t/q) as the flow rate at which the incumbent monopolist is replaced by an entrant. The definition of a BGP tells us that the interest rate r(t) and the innovation rate i(v,t/q) are both constant. Let us denote these two constants by r B and iB respectively. Then, we can rewrite the value function in Eq. (3) as V ðv; t=qÞ ¼ V ðqÞ ¼

βqR r B þ iB

:

ð4Þ

The expression r B + iB in the denominator on the right-hand-side (RHS) of Eq. (4) can be thought of as the effective discount rate in which the normal discount rate r B is adjusted because incumbent monopolists know that they will be replaced at the innovation flow rate iB. Let us now discuss the free entry condition in the R&D sector in our creative region. The innovation possibilities frontier in our creative region tells us that by spending one unit of the final consumption good, a firm generates—also see Section 2.2—a flow rate of innovation given by η/q(v,t). Because the improvement in quality in our model is probabilistic, the benefit from innovation is also stochastic. In the economy of our creative region, an apposite goal for potential entrants in the R&D sector is to maximize their expected value. In this regard, observe that the value of an innovation of quality α in a R&D sector in which the current quality is q is V(αq). We know from the discussion in Section 2.1 that the cumulative distribution function of α is G(α) and that this function is defined on a support given by the interval ^ Using this information, the expected value (EV) of a ½ð1−βÞ−ð1−βÞ=β ; α. firm, conditional on generating a successful innovation, is

227

Using Eq. (5), the condition governing free entry into the R&D sector of our creative region can be written as Zα^

η q

V ðαqÞdGðα Þ ¼ ð1−βÞ−ð1−βÞ=β

Zα^

η q

αβqR

ð1−βÞ−ð1−βÞ=β

r B þ iB

dGðα Þ ¼ 1;

where we have used Eq. (4) to simplify the first integral on the lefthand-side (LHS) of Eq. (6). To simplify Eq. (6) further, let the mean or average quality improvement from innovations in our creative region ^ α

be E½α ¼ ∫ ð1−βÞ−ð1−βÞ=β αdGðαÞ; where E[·] is the expectation operator. Then, using this expectation, we can write the free entry into R&D condition succinctly as βηΕ½α R r B þ iB

¼ 1:

ð7Þ

The first order necessary condition, i.e., the Euler equation, for an optimum to the representative creative class household's utility maximization problem is C_ ðt Þ r B −ρ ¼ gB : ¼ θ C ðt Þ

ð8Þ

Observe that we can also express our creative region's growth rate g B in terms of the R&D expenditures undertaken by entrants attempting to invent and produce new machines. From the discussion in the second paragraph of Appendix A, it follows that the output of the final consumption good O(t) is proportional to the mean total quality of the machines which we denote by Q(t). So, mathematically, we have Q(t) = ∫10 q(v, t)dv. Now, to find a closed-form or analytic expression for our creative region's long run growth rate gB in terms of the constants of the problem, we need to derive an alternate expression for Q(t). Since we have denoted the endogenous rate (and probability) of an innovation by iB, it follows that in a small time interval of length Δt, there will be a number iBΔt of entrants who manage to come up with an innovation. Let us denote the random set of machine lines that experience an innovation by Ω where Ω is a subset of the unit interval [0,1]. Then, the reader ought to understand that the complementary set ΩC describes the set of machine lines that do not experience an innovation. With the information in the preceding paragraph, we can write Z

Z

Z

Q ðt þ Δt Þ ¼

qðv; t þ Δt Þdv ¼

Ω

qðv; t þ Δt Þdv þ

ΩC

qðv; t þ Δt Þdv: ð9Þ

Because ∫ ΩC qðv; t þ ΔtÞdv ¼ ∫ ΩC qðv; tÞdv; the RHS of Eq. (9) can be simplified to yield Z Q ðt þ Δt Þ ¼

Ω

  qðv; t þ Δt Þdv þ 1−iB Δt Q ðt Þ þ oðΔt Þ;

V ðαqÞdGðα Þ: −ð1−βÞ=β

ð1−βÞ

ð10Þ

where, for the last function o(Δt), we have limΔt → 0o(Δt)/Δt = 0. Now, using the expectation operator, the notion of a conditional expectation,7 and the fact that quality improvements resulting from innovations and the innovations themselves are probabilistic, we can express the first term in the RHS of Eq. (10) as Z Ω

  qðv; t þ Δt Þdv ¼ E½aqðv; t Þ=Innovate ¼ E E a=qðv; t Þ; Innovateqðv; t Þ=Innovate:

Zα^ EV ¼

ð6Þ

ð5Þ 7

See Ross (1996, pp. 20–21) for additional details on these concepts.

ð11Þ

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Innovations that improve the quality of an existing machine and the innovations themselves arise independently of the present quality of machines. This tells us that E[α/q(v,t), Innovate] = E[α] and E[q(v, t)/ Innovate] = ∫Ωq(v, t)dv = Q(t)iBΔt. Using these last two pieces of information, the RHS of Eq. (11) can be simplified to Z Ω

qðv; t þ Δt Þdv ¼ E½α Q ðt ÞiB Δt:

ð12Þ

Using Eqs. (11) and (12), we can express the LHS of Eq. (10) as   Q ðt þ Δt Þ ¼ E½α Q ðt ÞiB Δt þ 1−iB Δt Q ðt Þ þ oðΔt Þ:

ð13Þ

We now need an equation that connects the BGP growth rate of our creative region or g B to the innovation flow rate iB. To derive this equation, note first that on the BGP, we must have gB ¼ Q_ ðtÞ=Q ðtÞ. Using

_ Eq. (13), this ratio QðtÞ=QðtÞ can be expressed as   Q ðt þ Δt Þ−Q ðt Þ 1 Q_ ðt Þ ¼ limΔt→0 Δt Q ðt Þ Q ðt Þ oðΔt Þ 1 B : ¼ ðE½α −1Þi þ limΔt→0 Δt Q ðt Þ

ð14Þ

Since limΔt → 0(Δt)/Δt = 0, Eq. (14) tells us that Q_ ðt Þ ¼ ðE½α −1ÞiB ¼ g B : Q ðt Þ

ð15Þ

Now, combining the results in Eqs. (7), (8), and (15), we obtain the closed-form expression for the BGP growth rate gB that we seek. The pertinent expression is gB ¼

βηΕ½α R−ρ θ þ ðE½α −1Þ−1

:

ð16Þ

Two aspects of Eq. (16) are worth highlighting. First, in the initial paragraph of Section 1.1, we pointed out that a key claim of Richard Florida is that the creative capital possessing members of the creative class are the basic driver of economic growth. Eq. (16) provides theoretical justification for this contention. Specifically, this equation tells us that the creative capital input R positively influences the BGP growth rate of our creative region. Put differently, all else being equal, a region with more creative capital will grow faster than a comparable region with less creative capital. Second, from Eq. (16), it is clear that for our creative region to grow at a positive rate, the inequality βηE[α]R N ρ must be satisfied. Our last task in this section is to specify the transversality condition. This condition tells us that the interest rate must be greater than the growth rate of our creative region. In symbols, we need rB N gB. Using Eqs. (8) and (16) the required inequality can be expressed as ρ N (1–θ)(E[α]-1)βηR. Using this inequality and the inequality stated at the end of the previous paragraph, we conclude that a BGP equilibrium with positive growth exists in our creative region as long as βηΕ½α R N ρ N ð1−θÞðE½α −1ÞβηR:

ð17Þ

This completes our analysis of the BGP equilibrium and the growth rate in our creative region. We now discuss why the lower limit of the support of α, the random variable that delineates the probabilistic or random effect of an innovation, takes the value specified in Section 2.1. 4. The value of the lower support Recall from Section 2.1 that an innovation gives rise to a probabilistic improvement of α over the previous technology in use and that G(α)

denotes the cumulative distribution function of α with support ^ So, the lower limit of the support of α is (1-β)-(1-β)/β. [(1–β)-(1-β)/β, α]. To understand why the lower limit of the support takes this precise value, we need to recognize that in the model of this paper, when it comes to the invention and production of machines, there is competition between firms that have access to different vintages of a particular machine variety. This means that when it comes to analyzing the outcome of innovations, we need to consider two cases. In the first case, an innovation is drastic in the sense that it corresponds to a sufficiently high value of α. When this happens, the firm responsible for this innovation can charge the unconstrained monopoly price which is essentially a markup over the marginal cost of production. By doing this, the relevant firm captures the entire market for the machines it produces. The significance of the lower limit of the support (1-β)-(1-β)/β is that it corresponds to the cutoff point for making an innovation drastic. In symbols, by ensuring that α ≥ (1-β)-(1-β)/β, we, in effect, are ensuring that no matter what the actual realization of α, the pertinent firm will be able to charge the unconstrained monopoly price and capture the entire market for the machines of this variety. If, on the other hand, we allow the possibility that α b (1-β)-(1-β)/β, then the value function in Eq. (4) ceases to be the correct value function because this value function assumes that the innovations are drastic and that the relevant firms are able to charge the unconstrained monopoly price. Specifically, if α b (1-β)-(1-β)/β then when an innovation does occur, the old incumbent firm producing the machine with quality (1/α)q(v,t) will be able to set a price for its machine that is low enough so that the competitive producers of the final consumption good will prefer the “old” quality at this low price. This, in turn, will obligate an entrant with a new but non-drastic innovation to engage in limit pricing.8 This involves setting a price—essentially the marginal cost of producing the machine of the “old” quality—so that the competitive producers of the final consumption good are indifferent between purchasing machines of the “old” quality and the machines of new but non-drastic quality. Even though this second case involving limit pricing is theoretically relevant, for the case of knowledge goods in creative regions that we are studying, we maintain that this case is of little practical interest. Therefore, we do not consider this case in the present paper. This course of action explains why the lower limit of the support of α takes the value (1-β)-(1-β)/β. Our next task is to state and solve the social planner's maximization problem and then derive the Pareto optimal growth rate in our creative region. 5. The socially planned equilibrium and growth rate We begin by procuring an expression for the net output that can be distributed between consumption and R&D in our creative region. To this end, let us use the superscript P to denote the Pareto optimal values of the pertinent variables. From the discussion in the third paragraph of Appendix A, it follows that the expression we seek is given by OP ðt Þ−X P ðt Þ ¼ βð1−βÞ−1=β Q P ðt ÞR ¼ IP ðt Þ þ C P ðt Þ:

ð18Þ

Now, to derive a differential equation for the mean total quality or QP(t), observe that R&D spending of IP(t) leads to the discovery of better vintages of machine varieties at the flow rate of η. In addition, each new vintage raises mean total quality by the proportional amount (E[α]-1). Using this piece of information and Eqs. (14) and (15), we infer that the differential equation we seek is given by P Q_ ðt Þ ¼ ηðE½α −1ÞIP ðt Þ:

8

See Acemoglu (2009, pp. 418–420) for a textbook account of these concepts.

ð19Þ

A.A. Batabyal, H. Beladi / Economic Modelling 53 (2016) 224–230

We are now in a position to state our social planner's dynamic or intertemporal maximization problem. Specifically, this planner solves Z∞ maxfC P ðtÞ;Q P ðt Þg∞ t¼0

e−ρt

C P ðt Þ1−θ −1 dt; 1−θ

ð20Þ

0

subject to the differential equation constraint n o P Q_ ðt Þ ¼ ηðE½α −1Þ βð1−βÞ−1=β Q P ðt ÞR−C P ðt Þ ;

ð21Þ

where we have used Eq. (18) to substitute for IP(t) in Eq. (19). The so called current value Hamiltonian function for the problem in (20)–(21) is n o C P ðt Þ1−θ −1 þ γP ðt Þ H Q P ðt Þ; C P ðt Þ; γP ¼ h 1−θ n oi  ηðE½α −1Þ βð1−βÞ−1=β Q P ðt ÞR−C P ðt Þ ; ð22Þ where γP(t) is the costate or adjoint variable associated with the constraint in Eq. (21). The two first order necessary and sufficient conditions for an optimum along with the transversality condition are ∂H ð; ; Þ ∂C P ðt Þ ∂H ð; ; Þ ∂Q P ðt Þ

¼ C P ðt Þ−θ −γP ðt ÞηðE½α −1Þ ¼ 0;

ð23Þ

¼ ργP ðt Þ−γ_ ðt Þ ¼ γP ðt ÞηðE½α −1Þβð1−βÞ−1=β R;

ð24Þ

P

and n o limt→∞ e−ρt γP ðt ÞQ P ðt Þ ¼ 0:

ð25Þ

Manipulating Eq. (24), we get βð1−βÞ−1=β ηðE½α −1ÞR−ρ ¼ −

P γ_ ðt Þ : γP ðt Þ

ð26Þ

Rewriting Eq. (23) and then using the result in Eq. (26), we obtain an expression for the Pareto optimal growth rate gP of our creative region and that expression is P C_ ðt Þ P

C ðt Þ

¼−

1 γ_ P ðt Þ βð1−βÞ−1=β ηðE½α −1ÞR−ρ ¼ ¼ gP : θ γP ðt Þ θ

ð27Þ

Inspecting Eq. (27), we see that as in Section 3, the creative capital input R once again positively affects the Pareto optimal growth rate in our creative region. We are now in a position to carry out the final task of this paper. This involves comparing the BGP (gB) and the Pareto optimal (gP)growth rates and then discussing when there is either too much or too little innovation in our creative region.

229

creative region is always different than the BGP growth rate. The reason why this comparison yields an equivocal result stems from the two effects that are generally present in Schumpeterian models of economic growth. As explained in Section 2.2, one effect is the profit stealing effect and the other effect is the appropriability effect. Recall that the profit stealing effect tells us that new entrants do not account for the effect stemming from the fact that they are replacing and hence stealing the profit of an existing monopolist producer of machines. This has the effect of raising the economic growth rate in our creative region. In contrast, the outcome of the appropriability effect is that the monopolistic machine producers are not able to capture or appropriate the entire benefit of an innovation. This has the effect of lowering the economic growth rate in our creative region. Since these two effects work in opposite directions, the net effect on the growth rate of the economy of our creative region is equivocal. Since the economic growth rate of our creative region is a positive function of innovations, this last result also tells us that it is not possible to unequivocally determine whether there is excessive innovation in our creative region in the BGP equilibrium. The above general point notwithstanding, all else being equal, the larger—or, in the language of Section 4, the more drastic—the mean quality improvement resulting from innovations or E[α], the more likely it is that the growth rate of our creative region with social planning will surpass its growth rate in the BGP equilibrium. Now, given that economic growth is a positive function of innovations, this means that the bigger is E[α], the more likely it is that there will be insufficient innovations taking place in the BGP equilibrium in our creative region. To see this, consider the two expressions for gB and gP in Eq. (28). We see that in the limit as E[α] → ∞, the ratio gP/gB → (1/β)- 1/β N 1 Hence, in this limiting case, the Pareto optimal growth rate is higher than the BGP growth rate and thus, from the perspective of both innovations and economic growth, our creative region will be better off with social planning. Looked at a little differently, the occurrence of drastic innovations provides a possible rationale for regulatory intervention in the economy of our creative region. Before we bring this sixth section to a close, we note that in our paper, we extend and thereby generalize the analysis in Acemoglu (2009, pp. 459–472) in four ways. First, in Acemoglu's model, conditional on success, an innovation leads to a deterministic improvement in quality over the previous technology. In contrast, in our model, conditional on success, an innovation results in a probabilistic improvement in quality of α over the previous technology. Second, we work with creative capital in our model and not, as Acemoglu does, with labor. In this regard, the reader should note that creative capital and labor are dissimilar inputs. Third and once again in contrast with Acemoglu's model, Section 2.1 points out that there is a spatial dimension in our analysis. Finally, the variables that affect the comparison between the two growth rates gB and gP—see (28)—in our model are different than the variables in Acemoglu's model. This concludes our study of the effects of probabilistic innovations on Schumpeterian economic growth in a creative region.

6. The BGP versus the Pareto optimal growth rates

7. Conclusions

The BGP and the Pareto optimal growth rates are given in Eqs. (16) and (27). Putting the two expressions side by side, we want to know whether

In this paper, we analyzed the effects of probabilistic or random innovations on Schumpeterian economic growth in a region that was creative in the sense of Richard Florida. Our analysis led to four findings. First, we described the BGP equilibrium and then computed the BGP growth rate in our creative region. Second, we explained why the lower limit of the support of the random variable that delineated the effect of an innovation, took the value that it did. Third, we solved the social planner's problem and derived the Pareto optimal growth rate in our creative region. Fourth, we compared the BGP and the Pareto

gB ¼

βηΕ½α R−ρ θ þ ðE½α −1Þ−1

≤ð≥Þ

βð1−βÞ−1=β ηðE½α −1ÞR−ρ ¼ gP : θ

ð28Þ

Comparing the above two expressions for gB and gP, the only thing we can say for sure is that the Pareto optimal growth rate of our

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A.A. Batabyal, H. Beladi / Economic Modelling 53 (2016) 224–230

optimal growth rates and then discussed when there was either too much or too little innovation in our creative region.9 The analysis in this paper can be extended in a number of different directions. Here are three suggestions for generalizing the research described here. First, an interesting extension that would also yield insights into phenomena occurring over space involves the analysis of a multi-region model to ascertain what kinds of spatial interactions between different creative regions can be studied theoretically. Second, it would also be useful to study the impact that the taxation of R&D by new entrants has on incumbent firms specifically and on economic growth in creative regions more generally. Finally, some innovations might prompt gradual changes in a creative region whereas others might give rise to sharp changes. This could be studied using the perspectives of complexity science. Studies that incorporate these aspects of the problem into the analysis will increase our understanding of the connections between probabilistic innovations and Schumpeterian economic growth in one or more creative regions. Appendix A Upon adapting equation 14.8 in Acemoglu (2009, p. 462) to our problem, we see that the equilibrium profit of a machine producer with quality q(v,t) is given by π(v,t/q) = βq(v,t)R. Similarly, adapting equation 14.13 in Acemoglu (2009, p. 462) to our problem, we deduce that the value function V(v,t/q) described in the first paragraph of Section 3 solves a so called Hamilton–Jacobi–Bellman equation _ given by rðtÞVðv; t=qÞ−Vðv; t=qÞ ¼ πðv; t=qÞ−iðv; t=qÞVðv; t=qÞ; where

_ Vðv; t=qÞ is the time derivative of the V(v,t/q) function. Adapting the discussion in Acemoglu (2009, p. 462) to our problem, it follows that the output of the final consumption good O(t) is proportional to the mean total quality of the machines which we denote by Q(t). Adapting equation 14.25 in Acemoglu (2009, p. 466) to our problem and using the superscript P to denote the Pareto optimal values of the pertinent variables, we get an equation for the net output that can be spread between consumption and R&D in the region under study. That equation is OP(t) − XP(t) = β(1 − β)−1/βQP(t)R = IP(t) + CP(t). References Acemoglu, D., 2009. Introduction to Modern Economic Growth. Princeton University Press, Princeton, NJ. Acs, Z., Audretsch, D., 1988. Innovation in large and small firms: an empirical analysis. Am. Econ. Rev. 78, 678–690. Aghion, P., Howitt, P., 1992. A model of growth through creative destruction. Econometrica 60, 323–351. Aghion, P., Blundell, R., Griffith, R., Howitt, P., Prantl, S., 2009. The effects of entry on incumbent innovation and productivity. Rev. Econ. Stat. 91, 20–32.

9 In this paper we conducted a theoretical analysis of the effects of probabilistic innovations on Schumpeterian economic growth in a creative region. To conduct an empirical analysis of this kind of question, we would first need appropriate data and then an estimation strategy. Speaking in general, as far as data are concerned, Schumpeterian growth models have often been estimated using micro firm level datasets. In this regard, firm level panel datasets describing the activities of firms in different nations have often been used by researchers. One key question here concerns the measurement of innovation. There are a variety of approaches available to researchers. For instance, following Acs and Audretsch (1988), we could measure innovative activity with the number of innovations in each four-digit standard industrial classification (SIC) industry. This kind of data has been collected by the United States Small Business Administration. Alternately, we could follow Lentz and Mortensen (2008) and use, for instance, Danish firm data—an annual panel of privately owned firms—that provide information on the connections between productivity, employment, and sales. As far as an estimation strategy is concerned, if we follow an approach of the sort used by Acs and Audretsch (1988), then we would use cross-section regression analysis to study the postulated logarithmic relationship between innovative output in a particular time period and innovation inducing inputs in the previous time period. Alternately, if we were to follow the Lentz and Mortensen (2008) approach then, using the equilibrium relationships of the pertinent model, we would estimate the parameters with a method of indirect inference that is based on information on valued added, employment, and wage payments from an apposite panel of firms. See Saunoris and Payne (2011) and the other studies discussed in Section 2.2 for more on empirical issues involving datasets and estimation strategies.

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